Invariants
| Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $576$ | ||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $4$ are rational) | Cusp widths | $1^{2}\cdot2\cdot3^{2}\cdot6\cdot8\cdot24$ | Cusp orbits | $1^{4}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 24G1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.96.1.1582 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}7&6\\4&23\end{bmatrix}$, $\begin{bmatrix}11&12\\12&19\end{bmatrix}$, $\begin{bmatrix}13&3\\20&11\end{bmatrix}$, $\begin{bmatrix}19&0\\8&23\end{bmatrix}$, $\begin{bmatrix}23&0\\16&17\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | Group 768.1035916 |
| Contains $-I$: | no $\quad$ (see 24.48.1.is.1 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $2$ |
| Cyclic 24-torsion field degree: | $16$ |
| Full 24-torsion field degree: | $768$ |
Jacobian
| Conductor: | $2^{6}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 576.2.a.b |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 156x + 560 $ |
Rational points
This modular curve has infinitely many rational points, including 4 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2^4\cdot3^4}\cdot\frac{96x^{2}y^{14}-4004640x^{2}y^{12}z^{2}+90114757632x^{2}y^{10}z^{4}-1061421137187840x^{2}y^{8}z^{6}+6488468676279336960x^{2}y^{6}z^{8}-21496755283181662961664x^{2}y^{4}z^{10}+32421467787354839596400640x^{2}y^{2}z^{12}-17388594333944875364414128128x^{2}z^{14}-4368xy^{14}z+144011520xy^{12}z^{3}-2312977358592xy^{10}z^{5}+22667751857590272xy^{8}z^{7}-126284010094627651584xy^{6}z^{9}+367213751802934056714240xy^{4}z^{11}-494977539190102077012443136xy^{2}z^{13}+243746275109686767350427156480xz^{15}-y^{16}+124800y^{14}z^{2}-3750831360y^{12}z^{4}+47155555123200y^{10}z^{6}-331495755120525312y^{8}z^{8}+1390496073211098169344y^{6}z^{10}-2794938739436762757070848y^{4}z^{12}+2430032602145395360017678336y^{2}z^{14}-698603337110790098828201558016z^{16}}{z^{2}y^{2}(x^{2}y^{10}-78624x^{2}y^{8}z^{2}+401381568x^{2}y^{6}z^{4}-447046594560x^{2}y^{4}z^{6}-278628139008x^{2}y^{2}z^{8}-30091839012864x^{2}z^{10}-80xy^{10}z+1804464xy^{8}z^{3}-6664902912xy^{6}z^{5}+6239303147520xy^{4}z^{7}-2995252494336xy^{2}z^{9}-300918390128640xz^{11}+2968y^{10}z^{2}-25104384y^{8}z^{4}+44948017152y^{6}z^{6}-17845100347392y^{4}z^{8}+11423753699328y^{2}z^{10}+1685142984720384z^{12})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.48.0-12.g.1.3 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.24.0-24.y.1.7 | $24$ | $4$ | $4$ | $0$ | $0$ | full Jacobian |
| 24.48.0-12.g.1.12 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.192.1-24.dh.1.14 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.192.1-24.dh.2.14 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.192.1-24.dh.3.16 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.192.1-24.dh.4.16 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.192.1-24.dj.1.12 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.192.1-24.dj.2.12 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.192.1-24.dj.3.16 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.192.1-24.dj.4.16 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.192.3-24.cg.1.37 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.192.3-24.cp.1.4 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.192.3-24.ds.1.7 | $24$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 24.192.3-24.du.1.6 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.192.3-24.fc.1.8 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.192.3-24.ff.1.10 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.192.3-24.ft.1.8 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.192.3-24.fu.1.4 | $24$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 24.192.3-24.go.1.8 | $24$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 24.192.3-24.go.2.8 | $24$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 24.192.3-24.go.3.16 | $24$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 24.192.3-24.go.4.16 | $24$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 24.192.3-24.gq.1.8 | $24$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 24.192.3-24.gq.2.8 | $24$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 24.192.3-24.gq.3.16 | $24$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 24.192.3-24.gq.4.16 | $24$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 24.288.5-24.ev.1.24 | $24$ | $3$ | $3$ | $5$ | $2$ | $1^{4}$ |
| 72.288.5-72.bm.1.28 | $72$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 72.288.9-72.da.1.28 | $72$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 72.288.9-72.di.1.30 | $72$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 120.192.1-120.sh.1.32 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.sh.2.30 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.sh.3.32 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.sh.4.30 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.sj.1.32 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.sj.2.28 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.sj.3.32 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.sj.4.28 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.3-120.ok.1.24 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.om.1.14 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.oo.1.6 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.oq.1.22 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.pq.1.30 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ps.1.22 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.pu.1.24 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.pw.1.14 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.se.1.16 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.se.2.16 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.se.3.32 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.se.4.32 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.sg.1.16 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.sg.2.16 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.sg.3.32 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.sg.4.32 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.480.17-120.brc.1.28 | $120$ | $5$ | $5$ | $17$ | $?$ | not computed |
| 168.192.1-168.sf.1.31 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.sf.2.31 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.sf.3.31 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.sf.4.31 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.sh.1.31 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.sh.2.31 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.sh.3.31 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.sh.4.31 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.3-168.lw.1.27 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.ly.1.30 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.ma.1.15 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.mc.1.14 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.nc.1.27 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.ne.1.22 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.ng.1.14 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.ni.1.14 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.pq.1.31 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.pq.2.31 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.pq.3.31 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.pq.4.31 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.ps.1.31 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.ps.2.31 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.ps.3.31 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.ps.4.31 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.1-264.sf.1.28 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.sf.2.28 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.sf.3.32 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.sf.4.32 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.sh.1.28 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.sh.2.28 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.sh.3.32 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.sh.4.32 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.3-264.lw.1.24 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.ly.1.20 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.ma.1.15 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.mc.1.20 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.nc.1.15 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.ne.1.26 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.ng.1.16 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.ni.1.20 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.pq.1.16 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.pq.2.16 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.pq.3.32 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.pq.4.32 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.ps.1.24 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.ps.2.24 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.ps.3.32 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.ps.4.32 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.1-312.sh.1.32 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.sh.2.32 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.sh.3.32 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.sh.4.32 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.sj.1.32 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.sj.2.32 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.sj.3.32 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.sj.4.32 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.3-312.ok.1.26 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.om.1.26 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.oo.1.14 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.oq.1.26 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.pq.1.26 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.ps.1.26 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.pu.1.30 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.pw.1.22 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.se.1.32 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.se.2.32 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.se.3.32 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.se.4.32 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.sg.1.32 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.sg.2.32 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.sg.3.32 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.sg.4.32 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |